Graphene - Green's function technique

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Discussion Overview

The discussion revolves around the application of Matsubara Green's function technique to graphene, focusing on the complexities arising from its two sublattices (A and B) and the implications for calculating the screened Coulomb potential and polarization bubble. Participants seek clarification and deeper understanding of the mathematical expressions involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the use of different Green's functions (G_{AA}, G_{AB}, G_{BA}, G_{BB}) in the context of graphene and their roles in evaluating the polarization bubble.
  • Another participant suggests that considering the expression for the bubble in coordinate space may provide clarity.
  • A follow-up response questions the relevance of the suggestion and requests further explanation on how locality affects the calculation in graphene.
  • One participant notes that the electromagnetic field couples locally to the electrons, implying that the bubble involves an integral of the product of two Green's functions.
  • A reference to a specific review is provided as a potentially helpful resource for understanding the topic better.
  • A participant asks for clarification on specific equations from a referenced document, indicating a need for deeper insight into the mathematical formulation.
  • Another participant points to a specific section in a different resource that may help derive a more general expression related to the topic.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints and remains unresolved, with participants seeking clarification and further explanation on various aspects of the Green's function technique as applied to graphene.

Contextual Notes

Participants express uncertainty regarding the appropriate Green's functions to use in calculations, and there are references to specific sections in external materials that may contain relevant information, but no consensus is reached on the best approach.

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Graphene -- Green's function technique

Hi,

I am looking for a comprehensive review about using Matsubara Green's function technique for graphene (or at least some hints in the following problem). I have already learned some finite temperature Green's function technique, but only the basics.

What confuses me is that graphene has two sublattices (say A and B), and so (in principle) we have four non-interacting Green's functions: G_{AA}(k,\tau)=-\langle T_{\tau}a_k(\tau)a_k^{\dagger}(0)\rangle, ,

where a_k is the annihilation operator acting on the A sublattice. G_{AB}, G_{BA} and G_{BB} are defined in a similar way.

Of course, there are connections between them, but G_{AA} and G_{AB} are essentially different. Now when I am to compute e.g. the screened Coulomb potential, I do not know, which Green's function should be used to evaluate the polarization bubble.

Thank you for your help!
 
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I think you will find the answer you are looking for when you consider the expression for the bubble in coordinate space.
 


Dear DrDu,

thank you for your response, but I do not think, I understand how your suggestion helps me. Please explain it to me a bit more thoroughly.
 


I mean that the electromagnetic field couples locally to the electrons. Hence the bubble is some integral containing a product of two Greensfunctions G(x,x')G(x,x'). What consequences does locality have in the case of Graphene?
 


Dear tejas777,

This is a very nice review, thank you very much. Let me ask just one final question: can you explain, how comes
F_{s,s'}(p,q)

in eq. (2.12) and (2.13) ?
 


Look at section 6.2 (on page 19/23) in:

http://nanohub.org/resources/7436/download/Notes_on_low_field_transport_in_graphene.pdf

Now, the link contains a specific example. You can probably use this type of approach to derive a more general expression, one involving the ##s## and ##s'##. I may have read an actual journal article containing the rigorous analysis, but I cannot recall which one it was at the moment. If I am able to find that article I will post it here asap.
 

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